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# Exponents

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### Exponents

1. 1. Exponents
2. 2. Review: <ul><li>What do exponents tell us? </li></ul><ul><li>What does 5 3 mean? </li></ul><ul><li>5 3 =5  5  5 </li></ul>
3. 3. What is the “base”? What is the “exponent”? <ul><li>8 5 </li></ul>Base Exponent
4. 4. <ul><li>Write 6 3 in standard form. </li></ul><ul><li>6 3 = 6  6  6 = 216 </li></ul>
5. 5. Write 6 4 in standard form. <ul><li>6 4 = 6  6  6  6 </li></ul><ul><li>= 1,296 </li></ul>
6. 6. What happens if we multiply <ul><li>6 3  6 4 ? </li></ul><ul><li>6 3  6 4 = </li></ul><ul><li>(6  6  6)  (6  6  6  6) </li></ul>6 3 6 4
7. 7. <ul><li>So… </li></ul><ul><li>6 3  6 4 = 6 7 because we used 6 as a factor seven times! </li></ul><ul><li>Notice anything? </li></ul>
8. 8. Let’s try another example: <ul><li>What is 5 2  5 4 ? </li></ul><ul><li>5 2  5 4 = (5  5)  (5  5  5  5) </li></ul><ul><li>or 5 2  5 4 = 5 6 </li></ul>
9. 9. For MULTIPLICATION: <ul><li>x s  x t = x s + t </li></ul><ul><li>(For any number x and for integers s and t .) </li></ul>
10. 10. <ul><li>In other words, when multiplying two numbers in exponential form with the same base , add the exponents. </li></ul>
11. 11. <ul><li>Ex : 9 4  9 8 = ? </li></ul><ul><li>9 12 </li></ul>
12. 12. What happens if we divide? 7 5 7 3 <ul><li>7 5 = 7  7  7  7  7 </li></ul><ul><li>7 3 7  7  7 </li></ul>Cancel out any numbers that are in the numerator AND denominator!
13. 13. <ul><li>After we cancel, we are left with 7 2 . </li></ul><ul><li>So, 7 5 = 7 2 </li></ul><ul><li>7 3 </li></ul>
14. 14. For DIVISION: <ul><li>x s = </li></ul><ul><li>x t </li></ul><ul><li>(For any nonzero number x and for integers s and t .) </li></ul>x s - t
15. 15. <ul><li>In other words, when dividing two numbers in exponential form with the same base , subtract the exponents. </li></ul>
16. 16. <ul><li>Ex: 6 8 ÷ 6 5 = ? </li></ul><ul><li>6 3 </li></ul>
17. 17. What about (5 2 ) 3 ? <ul><li>(5 2 ) 3 </li></ul><ul><li>(5 2 )  (5 2 )  (5 2 ) </li></ul><ul><li>(5  5)  (5  5)  (5  5) </li></ul><ul><li>5 6 </li></ul>
18. 18. Raising a power to a power: <ul><li>(x s ) t = x s  t </li></ul><ul><li>(For any number x and for integers s and t .) </li></ul>
19. 19. <ul><li>In other words, when raising a number with an exponent to a power , multiply the exponents. </li></ul>
20. 20. <ul><li>Ex : </li></ul><ul><li>(5 3 ) 7 = ? </li></ul><ul><li>5 21 </li></ul>
21. 21. One important thing to remember… <ul><li>x 0 = 1 </li></ul><ul><li>( Any number to the zero power is equal to 1 !) </li></ul>
22. 22. Why??? <ul><li>For example… </li></ul><ul><li>3 2 ÷ 3 2 = 3 2 - 2 = 3 0 </li></ul><ul><li>3 2 = 3  3 = 9 = 1 </li></ul><ul><li>3 2 3  3 9 </li></ul>
23. 23. <ul><li>So, 3 0 = 1 </li></ul><ul><li>This works for any base number! </li></ul>
24. 24. Can we have negative exponents? <ul><li>Yes! Negative exponents represent the reciprocal. </li></ul><ul><li>The reciprocal of x is The reciprocal of 10 is </li></ul><ul><li>They do NOT make the answer negative. </li></ul>
25. 25. <ul><li>5 -2 = = </li></ul><ul><li>2 -4 = </li></ul>